document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92733>Question 92733</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the inequality: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5x%3C4 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5x%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's what i did: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5x%3C4 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5x%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "                   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5x-4%3C0 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5x-4%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "                   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%283x%2B4%29%3C0 ALIGN=MIDDLE  ALT=\"%282x-1%29%283x%2B4%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now what? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #67501 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=67501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Solve the inequality: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5x%3C4 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5x%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's what i did: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5x%3C4 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5x%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" > <===ok ... this is given<BR>\n" );
document.write( "                   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5x-4%3C0 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5x-4%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== Step 1 ok<BR>\n" );
document.write( "                   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%283x%2B4%29%3C0 ALIGN=MIDDLE  ALT=\"%282x-1%29%283x%2B4%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== Step 2 ok<BR>\n" );
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document.write( "What you've done so far is very good. You just need a little prompting to hopefully see<BR>\n" );
document.write( "the light at the end of the tunnel.<BR>\n" );
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document.write( "Let's back off Step 1 and Step 2 above and think about them more in the form of an equation.<BR>\n" );
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document.write( "In a general sense we can say that a quadratic equation is of the form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+y ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "From what you probably know about quadratic equations, you can tell that their graphs are<BR>\n" );
document.write( "parabolas. If \"a\" (the multiplier of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is positive the graph is loosely shaped like<BR>\n" );
document.write( "the letter \"U\" ... as you look horizontally from left to right the graph descends to<BR>\n" );
document.write( "a low point, and then turns and ascends towards infinity.<BR>\n" );
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document.write( "Now think of your Step 1 as being in the form of a quadratic equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5x-4+=+y ALIGN=MIDDLE  ALT=\"6x%5E2%2B5x-4+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Because the multiplier of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term is positive, the parabolic graph of this equation<BR>\n" );
document.write( "will be U-shaped. If the graph crosses the x-axis, the coordinates of the points where it <BR>\n" );
document.write( "crosses will have their y values equal to zero. So by setting y equal to zero we can find<BR>\n" );
document.write( "the values of x that correspond.  So set y equal to zero and solve the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5x-4+=+0 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5x-4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can now factor this equation just as you did in Step 2 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%283x%2B4%29=0 ALIGN=MIDDLE  ALT=\"%282x-1%29%283x%2B4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then, with this quadratic polynomial equal to zero the equation will be true if either of <BR>\n" );
document.write( "the two factors is equal to zero because a multiplication by zero on the left side makes <BR>\n" );
document.write( "the entire left side equal to the zero on the right side.<BR>\n" );
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document.write( "Therefore, if either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-1=0 ALIGN=MIDDLE  ALT=\"2x+-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4+=+0 ALIGN=MIDDLE  ALT=\"3x%2B4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > then the quadratic polynomial<BR>\n" );
document.write( "will equal zero and the values of x that make that happen will be the values on the x-axis<BR>\n" );
document.write( "where the graph intersects that axis. Solve these two equations and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-+1+=+0 ALIGN=MIDDLE  ALT=\"2x+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== Add 1 to both sides and then divide both sides by 2 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F2 ALIGN=MIDDLE  ALT=\"x+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3x+%2B+4+=+0 ALIGN=MIDDLE  ALT=\"+3x+%2B+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== Subtract 4 from both sides and then divide both sides by 3 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-4%2F3 ALIGN=MIDDLE  ALT=\"x+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This tells us that the graph intersects the x-axis at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-4%2F3 ALIGN=MIDDLE  ALT=\"x+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F2 ALIGN=MIDDLE  ALT=\"x+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "For values of x between those two points the graph is sagging below the x-axis. (For values of<BR>\n" );
document.write( "x greater than +1/2 or less than -4/3 the graph will be above the x-axis.)<BR>\n" );
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document.write( "Now recall that in Step 1 you had the problem down to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2B5x-4%3C0 ALIGN=MIDDLE  ALT=\"6x%5E2%2B5x-4%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You now know that as indicated by its graph, the polynomial on the left side will have<BR>\n" );
document.write( "a y value of less than zero when the graph is in the \"sagging\" portion between the two<BR>\n" );
document.write( "values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-4%2F3 ALIGN=MIDDLE  ALT=\"x+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F2 ALIGN=MIDDLE  ALT=\"x+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Any value of x between those two values will<BR>\n" );
document.write( "make the polynomial have a value less than zero as indicated by the fact that the y<BR>\n" );
document.write( "value (which equals the value of the polynomial and corresponds to the graph) will be <BR>\n" );
document.write( "negative and therefore less than zero.<BR>\n" );
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document.write( "We can write the answer to the problem as:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3+%3C+x+%3C+1%2F2 ALIGN=MIDDLE  ALT=\"-4%2F3+%3C+x+%3C+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and that's the answer you are looking for.  <BR>\n" );
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document.write( "Hope that something in this long discussion of the problem gets you thinking about a way<BR>\n" );
document.write( "to visualize what this problem means in relation to the graph and how the graph can help<BR>\n" );
document.write( "you to interpret the answer.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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